[1]} identify the loadings and idiosyncratic variations by
setting \(D=I\) , the upper triangular elements of \(\Lambda \)  to zero
and the leading diagonal elements to positive values \(\lambda _{i,i}>0\) .
The copula parameters are then
\(\text{$\theta $}=(\mbox{vecl}(\Lambda ),d_1,\ldots ,d_m)\) , where
\(\mbox{vecl}(\Lambda )\)  is the half-vectorization
operator applied to the lower triangle
of the rectangular matrix \(\Lambda \) .
In the non-copula factor model literature, there are alternative
ways to identify \(\Lambda \)  and \(D\)  [2]}, [3]}, and
similar restrictions may be adapted for the correlation matrix \(\Omega \)  as well.
In a Bayesian framework, priors also have to be adopted for \(\Lambda \)  and \(D\) ,
and these can be used to provide further regularization as in [1]}
and elsewhere.
